Skinny- two separate (but related) meanings in the song:

mean, miserly. Gaffer's too skinny to pay.

cf
Copper on the corner, eating apple pie,
I asked him for a skinny bit and he hit me in the eye

(kid's song, Salford, Lamcs 1950s).

Thin, meagre, underfed She used to be skinny, now look at the pinny..

cf
Skinny Lizzie is the 'arem queen
(Mike Harding song)